Live plants are not necessary to a healthy fish tank. That being said, they are lovely to look at, and are beneficial as they reduce nitrogenous waste levels (ammonia/nitrite/nitrates). They also add oxygen to the water column during the day.
Having live plants depends on a number of factors.
What type of fish are you considering? Some fish are not compatible with plants, either because of water parameters needed (rare) or because the fish will uproot and eat the plants (more common).
How much light will the tank have? A 20g tank should have a minimum of 40 watts (2 watts per gallon). I'm guessing your single bulb is a standard flourescent with 15 watts. That won't be enough for even low light plants
IMHO.
Most folks use bulbs with a temp range of 6500K to 10,000K, and have between 1 and 4 watts of light per gallon. High levels of light for plants also require the addition of
CO2 (the carbon is a necessary nutrient for the plants; higher lighting, faster growing plants, more nutrients needed). Fertilizers will also be a big plus.